Glossary

Matrix representation of a linear system

Consider the following linear system

$x_1 - 2x_2 +7x_3 = -1$

$-x_1 +3x_2 -2x_3= 3$

Observe that since we know the number of equations and variables, the important information about this system are the number multiplying the variables and those ones on the right hand side of each equation. Thus there is a more compact way of encoding all the information needed about a linear system. We encode all the coefficients into a matrix called the matrix of coefficients

$\begin{bmatrix} 1 & -2 & 7 \\ -1 & 3 & -2 \end{bmatrix}$

If we want to capture the whole information we encode it into the augmented matrix

$\begin{bmatrix} 1 & -2 & 7 & -1 \\ -1 & 3 & -2 & 3\end{bmatrix}$

Seeing a matrix as a linear system

The above representation goes both ways. Consider the matrix below

$\begin{bmatrix} 2 & -7 & 1 & 1 \\ 0 & 3 & -2 & 3 \\ 1 & 0 & -\pi & 0 \end{bmatrix}$

it represents the following linear system

$2x_1 - 7x_2 +x_3 = 1$

$\quad \;\;- 3x_2 -2x_3 = 3$

$x_1 \quad \;\; -\pi x_3 = 0$